bioinformatics - University of Colorado Colorado Springs

BIOINFORMATICS
Vol. 00 no. 00 20xx
Pages 1–9
A Multiobjective Memetic Algorithm for PPI Network
Alignment
Connor Clark 1, and Jugal Kalita 1
Department of Computer Science, University of Colorado Colorado Springs, 1420 Austin Bluffs
Parkway, Colorado Springs, CO 80918, USA
1
Received on XXXXX; revised on XXXXX; accepted on XXXXX
Associate Editor: XXXXXXX
ABSTRACT
Motivation: There recently has been great interest in aligning proteinprotein interaction (PPI) networks in order to identify potentially
orthologous proteins between species. It is thought that the
topological information contained in these networks will yield better
orthology predictions than sequence similarity alone. Recent work
has found that existing aligners have difficulty making use of both
topological and sequence similarity when aligning, with either one or
the other being better matched. This can be at least partially attributed
to the fact that existing aligners try to combine these two potentially
conflicting objectives into a single objective.
Results: We present Optnetalign, a multiobjective memetic algorithm
for the problem of PPI network alignment that uses swap-based local
search, mutation, and crossover to create a population of alignments.
This algorithm optimizes the conflicting goals of topological and
sequence similarity separately, exploring the tradeoff between the
two objectives as it runs. This allows us to produce many highquality candidate alignments in a single run. Our algorithm produces
alignments that are much better compromises between topological
and biological match quality than previous work, while better
characterizing the diversity of possible good alignments between two
networks. Our aligner’s results have several interesting implications
for alignment evaluation, the design of network alignment objectives,
and the interpretation of alignment results.
Availability: The C++ source code to our program, along with
compilation and usage instructions, is available at
https://github.com/crclark/optnetaligncpp/
Contact: cclark@uccs.edu, jkalita@uccs.edu
1
INTRODUCTION
As the sizes of known protein-protein interaction (PPI) networks
grow, so does interest in analyzing them. One of the more ambitious
efforts in this area is aligning the PPI networks of two different
species, with the goal of identifying orthologous proteins as well
as shared pathways and complexes that hint at the PPI network
of a common ancestor. There has already been a good deal of
progress in this area, with past work reporting success in finding
large shared subnetworks between S. cerevisiae and H. sapiens,
as well as success in reconstructing phylogeny based on network
overlap discovered by an aligner (Kuchaiev et al., 2010; Kuchaiev
and Przulj, 2011).
c Oxford University Press 2014.
Despite these successes, PPI network alignment is a young
research area. The incomplete, noisy nature of existing PPI networks
makes alignment difficult, with existing aligners performing
dramatically better on noise-free synthetic networks compared to
noisy real-world networks (Clark and Kalita, 2014). Furthermore, it
has been found that the two objectives of biological and topological
fit that these aligners optimize conflict to a larger extent than
previously realized. Different aligners construct alignments that
are dramatically different, with some optimizing topological fit at
the expense of sequence similarity, while others do the opposite.
While some of these aligners provide a user-controlled parameter
for adjusting between optimizing one objective over the other, even
when this parameter is adjusted as widely as possible, existing
aligners only output alignments in small, non-overlapping regions
of objective space that are often quite distant from each other (Patro
and Kingsford, 2012; Clark and Kalita, 2014). Lastly, these aligners
tend to produce very different alignments, sometimes agreeing on as
few as 5% of their total aligned pairs (Patro and Kingsford, 2012),
which makes it necessary for the user to run many different aligners
to get an idea of the many alignments possible. The difficulties with
the conflict between topological and biological fit, and the disparate
results of existing aligners, show that the problem of PPI network
alignment is far from solved.
To this end, we introduce a multiobjective memetic algorithm,
Optnetalign (Optimizing Network Aligner), for performing network
alignment. Unlike most existing aligners which use custom-built
estimators of node similarity which don’t always have a clear
connection to final alignment quality, we use a multiobjective
memetic algorithm (Deb, 2001), which can produce a large number
of diverse, high quality alignments in a single run. Our approach
is able to better jointly optimize the topological and sequence
similarity of the proteins aligned, and instead of arbitrarily emitting
a single alignment at one point in objective space, Optnetalign
produces a Pareto front of alignments that characterize the wide
variety of possible tradeoffs between the conflicting objectives
of network alignment. This tends to produce a wider variety of
alignments than using all previously-created aligners together, and
these alignments tend to be comparable to or of better quality than
those produced by previous aligners.
1
C. Clark and J. Kalita
2
2.1
METHODS
Protein-Protein Interaction Networks and The Goal
of Alignment
A protein-protein interaction network is a graphical representation of all the
proteins in a given organism, where the nodes of the network represent
proteins, and a link between two proteins indicates that they interact to
perform some biological function. Protein interaction is measured through
a number of different experimental methods, with the most common being
coimmunoprecipitation (Aebersold and Mann, 2003). A large amount of
effort has gone into collecting and curating the results of many protein
interaction experiments, and the results are stored in databases such as
DIP (Xenarios et al., 2002), Isobase (Park et al., 2011), BIOGrid (Chatraryamontri et al., 2013), and STRING (Franceschini et al., 2013). A related
research effort in this area is the Gene Ontology (GO), a large database
annotating genes and proteins with information about their function, the
biological processes in which they participate, and the cellular components
in which they are found (Ashburner et al., 2000). The GO database is
constantly being updated and revised as new discoveries are made. An
important area of research is the transfer of GO annotations from proteins
in one species to orthologous proteins in another. Some species have more
complete GO annotation data than others, so it is very desirable to find a
way to accurately and automatically predict GO annotations for unannotated
proteins in less-studied species.
PPI network alignment is a way to use both the sequence similarity
between proteins, as well as the topology of PPI networks, to find likely
orthologous proteins between species. The goal is to create algorithms that
can automatically find likely subnetworks of two species’ PPI networks
that are common to the two species, and from there form hypotheses about
protein function in species for which less information exists. A single run of
an alignment program can generate thousands of candidate ortholog pairs.
2.2
Pairwise Global Alignment
We are given two graphs G1 = (V1 , E1 ) and G2 = (V2 , E2 ), whose
vertices represent proteins, and the presence of an edge (u, v) in E1 (or E2 )
indicates that the two proteins represented by u and v interact in G1 (or G2 ).
We assume, without loss of generality, that |V1 |  |V2 |.
The problem of pairwise global alignment is to find a one-to-one function1
f : V1 ! V2 that maps each node in V1 to the node it best matches
in V2 , according to the topology of the network around the node and the
sequence similarity of the proteins represented by the nodes. The various
methods for determining the “best” match will be described further below.
The resulting alignment f shows us a subnetwork that the two networks have
in common. The problem of global alignment is equivalent to the subgraph
isomorphism problem, which is NP-Complete (Cook, 1971), so aligners
produce approximate solutions using a variety of techniques that will be
discussed below.
2.3
Evaluating Alignment Quality
Once an alignment has been constructed, we must measure the extent to
which it has satisfied the two objectives of maximizing both the topological
fit between the two networks and the number of likely orthologs that have
been matched. These objectives are intended to measure the extent to
which we have identified overlapping pathways and orthology relationships
respectively. Three closely related metrics for topological fit have been used
in the global network alignment literature, which present different ways
of evaluating how well one network has been matched to the other: EC
(Kuchaiev et al., 2010), ICS (Patro and Kingsford, 2012), and S 3 (Saraph
and Milenković, 2014). Of these, EC is the most universally used, while
ICS and S 3 are newer refinements of EC. These three metrics are ratios
1
In the definitions that follow, we will employ a shorthand where f is also
applied to edges. In such cases f ((u, v)) is shorthand for (f (u), f (v)). This
makes several definitions significantly easier to read.
2
taking values between 0 and 1, with higher values being better. They all
share the same numerator, which represents the number of edges that have
been conserved by the alignment f .
conserved(f ) = |f (E1 ) \ E2 |
(1)
They differ, however, in their denominators. EC simply divides by
|E1 |, showing us how many of the edges in the first network have been
successfully matched. ICS refines this by instead dividing by |EG2 [f (V1 )] |,
the number of edges in the induced subgraph of G2 that f maps to. This
effectively penalizes alignments from sparse graph regions to dense graph
regions. S 3 is a further refinement of ICS, which penalizes dense-to-sparse
as well as sparse-to-dense alignments by dividing by |E1 | + |EG2 [f (V1 )] |
|f (E1 )|. While we previously evaluated existing algorithms using ICS
(Clark and Kalita, 2014), we switch to S 3 in this paper because, in contrast
to the aligners we evaluated previously, Optnetalign is able to “cheat”
when optimizing ICS by minimizing the denominator without increasing
the numerator, and MAGNA exhibits this behavior as well (Saraph and
Milenković, 2014). S 3 properly penalizes this behavior.
We also use a measure of agreement derived from biological knowledge.
The true alignment between two PPI networks is at least partially unknown,
so one cannot simply report the percentage of nodes mapped to their true
orthologs, except when using synthetic benchmark data. However, we can at
least verify that the proteins mapped to one another are similar in function.
All the literature on PPI network alignment makes use of GO annotations
to evaluate the accuracy of their alignments, by comparing the similarity of
GO annotations between aligned proteins. Several papers simply report the
percentage of aligned pairs that share more than k GO annotations (Kuchaiev
and Przulj, 2011; Memisević and Przulj, 2012; Neyshabur et al., 2013). We
use a Jaccard index, introduced by Aladag and Erten (2013). In the context
of network alignment, this measure is called GO consistency, and is defined
as
GOC(u, v) =
|GO(u) \ GO(v)|
|GO(u) [ GO(v)|
(2)
for an aligned pair of nodes u 2 V1 and v 2 V2 , where GO(u) denotes
the GO terms associated with the protein u. We restrict this to GO terms at
distance five from the root of the ontology, so as not to confuse the analysis
by using GO terms of differing specificity. We then report the sum of the GO
consistency over all alignment pairs for each alignment produced.
To further ease comparison with existing aligners, we evaluate on the
same dataset of Isobase networks as we used in our previous work (Clark
and Kalita, 2014) (for information on these networks, see our supplementary
information file). The only differences are that we now use the S 3 metric
instead of ICS, and, since the synthetic dataset NAPAbench (Sahraeian and
Yoon, 2012) did not exhibit the same tradeoff between ICS and biological
similarity that is exhibited in the real-world Isobase networks (Clark and
Kalita, 2014), we evaluate primarily on the Isobase networks, using the
NAPAbench data to verify the correctness of our alignments, since the true
alignments for NAPAbench are known.
Both topological similarity techniques and sequence similarity have
advantages and disadvantages. It has been argued that over-reliance on
topological similarity can be misleading, since actual complexes may
appear disconnected in current noisy, incomplete datasets, and so sequence
similarity information is essential to produce the best alignment possible
(Huang et al., 2012). Sequence similarity scores also have their problems,
since the actual level of sequence similarity between two proteins that
serve a similar function can vary (Chindelevitch et al., 2013). Previous
work has found (Patro and Kingsford, 2012; Clark and Kalita, 2014) that
existing aligners uncover a tradeoff between topological fit and biological fit
when constructing alignments– it is not possible to jointly maximize both.
This tradeoff can be quite dramatic. For example, on the C. elegans - D.
Melanogaster alignment problem, NETAL (Neyshabur et al., 2013) achieves
an S 3 of 0.41 and a GOC of 12, while PINALOG (Phan and Sternberg, 2012)
manages a S 3 of 0.065 and a GOC of 230. When existing aligners are all
plotted for one of these problems, we see a distinctive curve of tradeoffs
between these two objectives. We also find that these aligners produce
alignments in non-overlapping regions of the objective space, even for those
aligners that provide a parameter for controlling the tradeoff between the
two objectives. As we explain further below, ours is the only aligner that
was designed to explicitly explore the possible tradeoffs between these two
objectives, and to output many alignments for further analysis.
2.4
Existing Alignment Algorithms
One of the first algorithms for global network alignment was IsoRank (Singh
et al., 2008). This was then extended into a version that could align many
networks simultaneously, called IsoRankN (Liao et al., 2009). These two
work by computing a similarity measure similar to Google’s PageRank
algorithm, and this is combined with BLAST bit scores to find a mapping.
Another notable set of aligners is the GRAAL family (Kuchaiev et al., 2010;
Milenković et al., 2010; Kuchaiev and Przulj, 2011; Memisević and Przulj,
2012). These work by counting small induced subgraphs called “graphlets”
to compute pairwise node similarity. They differ mainly in what additional
similarity measures they use, and in their matching stages. NATALIE
2.0 frames alignment as a relaxation of an integer linear programming
problem (El-Kebir et al., 2011). GHOST uses methods from spectral graph
theory to estimate node similarity (Patro and Kingsford, 2012). PINALOG
recursively maps networks together by first matching dense subgraphs,
and then matching the nodes within those subgraphs (Phan and Sternberg,
2012). NETAL uses topological data exclusively, iteratively computing
match confidence as the expected number of conserved edges (Neyshabur
et al., 2013). SPINAL uses both coarse- and fine-grained matching steps
to iteratively improve an alignment (Aladag and Erten, 2013). PISwap
(Chindelevitch et al., 2013) first creates a maximum-weight match by
bitscore, and then improves the topological fit of the alignment through
swap-based local search. MAGNA (Saraph and Milenković, 2014) is a
topology-only aligner that uses a crossover-only genetic algorithm.
Most aligners can be categorized as belonging to one of two classes. The
first class, which has been much more popular, is what we call two-stage
alignment. Such aligners first compute some estimate of the similarity of
each pair of nodes in V1 ⇥ V2 using some topological similarity metric and
(in most cases) BLAST bit scores or E-values, and then use this similarity
matrix to create a maximum-weight bipartite matching between the nodes of
V1 and V2 . A simple strategy for this matching is the Hungarian algorithm,
which produces an optimal matching (Milenković et al., 2010; Chindelevitch
et al., 2013). However, because the similarity scores used are themselves
only approximate, the O(n3 ) time complexity of the Hungarian algorithm is
generally not worth the time, and most aligners favor faster, greedy matching
algorithms. Many aligners, such as GRAAL (Kuchaiev et al., 2010), MIGRAAL (Kuchaiev and Przulj, 2011), and GHOST (Patro and Kingsford,
2012), use variants on a “seed-and-extend” method, where the most similar
pair of nodes is aligned first, and then nodes neighboring that pair are
matched.
The second, and much smaller class, that is sometimes used in network
alignment are search-based aligners. These aligners use heuristic or
metaheuristic algorithms to continuously refine a single alignment or a
population of alignments. With such algorithms, the alignment objectives
that these aligners are trying to optimize are computed continuously as the
alignment is constructed, instead of being measured afterward as they are in
the two-stage aligners. Only a handful of search-based aligners exist. These
are SIM-T (Kpodjedo et al., 2014), which uses tabu search, where “moves”
are additions and removals of aligned pairs from the alignment; PISwap
(Chindelevitch et al., 2013), which uses the 3-opt heuristic to interchange
the assignments of three aligned pairs at each step; and MAGNA (Saraph and
Milenković, 2014), which uses a genetic algorithm that attempts to maximize
either EC, ICS, or S 3 . SPINAL (Aladag and Erten, 2013) also performs
some amount of swap-based hill climbing in its “fine-grained” matching
stage. Outside of the biological network alignment literature, the use of
metaheuristic search for graph matching has been researched as well, but
these approaches are less relevant as they only optimize topological fit and
are applied to graphs much smaller than those seen in bioinformatics (Cross
et al., 1997; Cicirello and Smith, 2000; Barecke and Detyniecki, 2007; Lipets
et al., 2009).
Optnetalign falls into the search-based category, because we believe the
two-stage paradigm has a number of severe weaknesses. First of all, twostage aligners typically introduce both a novel way of computing pairwise
node similarity along with a new matching algorithm. Little work exists on
mixing-and-matching approaches to the two stages to discover whether the
first or second stage is more responsible for the results of a given aligner.
To our knowledge, the only works attempting this compared only MIGRAAL to IsoRankN (Milenković et al., 2013) and MI-GRAAL to GHOST
(Crawford et al., 2014). Since two-stage algorithms do not track alignment
quality as they construct their alignments, it is very difficult to understand
what portions of a two-stage algorithm are responsible for the quality of the
results. This makes it difficult to create a new algorithm of this type. As
Saraph and Milenković (2014) recently observed, these two-stage aligners,
while hoping to optimize metrics such as EC and some metric of GO term
agreement, actually optimize ad-hoc similarity scores that have no clearly
established relation to standard alignment evaluation metrics. The existing
literature provides little explanation of why or whether such similarity scores
would be expected to work. In contrast, search-based aligners focus on
maximizing the actual alignment objectives that are currently accepted in
the alignment literature.
A problem shared by all existing aligners is that they do not explicitly
approach the problem of network alignment as a multiobjective one. It has
become increasingly clear that there is a tradeoff between our alignment
objectives, as we discussed above. However, the best any existing aligner
does to address this problem is providing users with a parameter ↵, which
gives the user rough control over the weight of biological or topological
similarity in constructing an alignment, with some equation similar to
sim(u, v) = ↵ ⇥ t(u, v) + (1
↵) ⇥ b(u, v)
(3)
where sim(u, v) is the aligner’s estimated overall similarity between nodes
u and v and t(u, v) and b(u, v) are the aligner’s estimated topological and
sequence similarity of the two nodes respectively. This is insufficient. Not
only does this approach requires trial-and-error experiments on the part of
the user, and complete recalculation of the alignment each time ↵ is adjusted,
but existing aligners only cover a narrow section of the objective space, even
when varying ↵ as widely as possible. Optnetalign produces a variety of
solutions more diverse than existing aligners combined, and does so in a
single run without extra effort on the part of the user.
2.5
Multiobjective Optimization
Metaheuristics are a family of optimization techniques for problems that
cannot be framed in a way amenable to standard optimization techniques,
such as when objective functions are discontinuous, not differentiable, or not
functions of real numbers. In PPI network alignment, our goal is to maximize
several objectives that are a function of a network alignment – the problem
must first be relaxed for standard optimization techniques to work (El-Kebir
et al., 2011). Metaheuristics are thus a natural fit for network alignment.
Genetic algorithms are a particular family of metaheuristics inspired by
biological evolution (Michalewicz and Fogel, 2004). Initially, a population
of candidate solutions are generated at random. These solutions are ranked
by their fitness, and the best of them are allowed to “breed” to produce a
new population of solutions. The new population undergoes mutation with
some small probability, and then the process begins again, until an acceptable
solution has been found (Floreano and Mattiussi, 2008). This approach is
applicable to many problems, since it only requires a way of scoring the
“fitness” of candidate solutions, and a data structure representing solutions
for which we can define a useful crossover operation (for combining
two existing solutions) and a mutation operation (for randomly modifying
solutions). Memetic algorithms are a generalization of standard genetic
algorithms that additionally incorporate a local search heuristic to further
3
C. Clark and J. Kalita
improve their results (Blum et al., 2011). Genetic algorithms and local search
heuristics are complementary – the former explores large spaces quickly
through crossover and mutation, while the latter is particularly effective at
refining solutions that have been found.
While genetic algorithms have been used since the 1970’s (e.g. Holland
(1975)), variants capable of efficiently optimizing multiobjective problems
have only become a focus of research more recently, with algorithms such
as NSGA-II (Deb et al., 2002) and SPEA2 (Zitzler et al., 2001) becoming
immensely popular. Other local search approaches to multiobjective
optimization have had great success, as well (Czyzżak and Jaszkiewicz,
1998; Corne et al., 2000; Knowles and Corne, 2000). Multiobjective
memetic algorithms have also been developed in recent years (Knowles
et al., 2001; Knowles and Corne, 2005). The key difference between single
objective metaheuristics such as MAGNA (Saraph and Milenković, 2014)
and multiobjective metaheuristics such as Optnetalign is that the latter
can separately optimize several objectives, even if they conflict, and then
output a representative set of solutions that are optimal tradeoffs between
the conflicting objectives. To accomplish this, multiobjective optimization
is framed in terms of several concepts borrowed from welfare economics
(Deb, 2001; Zhou et al., 2011). The first is Pareto optimality: a solution to
a multiobjective optimization problem is Pareto optimal if further increasing
one of the objectives would decrease one or more of the other objectives.
Such solutions represent optimal tradeoffs between the given objectives.
The second concept is Pareto dominance: solution x dominates solution
y if x is at least as good as y with respect to all objectives, and
strictly better than y with respect to at least one objective. We prefer a
solution x to a solution y if x Pareto dominates y, and we are indifferent
between x and y if neither dominates the other. This definition leads to
the possibility that a set of solutions may be non-dominated with respect
to each other. Multiobjective genetic algorithms work by selecting nondominated members of the population for reproduction at every generation,
in the hopes of approximating the set of Pareto optimal solutions. This
approximate Pareto optimal set will, in turn, allow us to characterize the
Pareto front of the objective space – the boundary between attainable and
unattainable solutions. Multiobjective genetic algorithms output a number
of representative solutions from the Pareto optimal set. Since these solutions
are all Pareto optimal, we cannot a priori prefer one of them over another,
and a human decision maker must use some other information or subjective
preference to decide which of these solutions to keep. In the case of
PPI network alignment, this means that our algorithm produces a number
of alignments with different tradeoffs between topological and biological
fit. These alignments can then be studied further to better understand the
relationship between the two networks under consideration.
2.6
Our Algorithm
We present a multiobjective memetic algorithm that discovers a
representative set of non-dominated alignments using crossover, mutation,
and swap-based hill climbing. Our algorithm maintains a population of
non-dominated alignments and attempts to improve them. The algorithm
initializes with randomly-constructed alignments. On each iteration,
Optnetalign selects random members of the population, performs swapbased crossover and mutation, followed by swap-based hill climbing. We
randomly select either hill climbing that improves one objective while
ignoring the others, or hill climbing that only makes moves that do
not worsen any objective. The resulting alignment is then placed in the
population if it is not Pareto dominated by any other population member,
and alignments it Pareto dominates are removed from the population. This
continues until a user-specified time limit is reached, after which the program
outputs all non-dominated solutions found. Generally, we find that the
algorithm converges after about 10 hours of runtime.
Following other genetic algorithms for network alignment, we adopt a
permutation encoding for our alignments (Barecke and Detyniecki, 2007;
Saraph and Milenković, 2014). In this encoding, we label the n nodes of each
network with the integers {0, 1, ...n 2, n 1}. Since we assume |V2 |
4
|V1 | , we add “dummy” nodes to V1 so that |V1 | = |V2 |. Then our alignment
f can be stored in memory as an array a such that if f (u) = v, then au =
v. All indices i
|V1 | of a are ignored. This representation simplifies
searching through the space of alignments to searching through the space
of permutations of integers in the interval [0, |V2 |), allowing us to adapt
permutation-based search operators that have been well-studied in genetic
and local search algorithms for the traveling salesman problem (Goldberg,
1989).
We adopt the Uniform Partially Matched Crossover (UPMX) operator
introduced in a genetic algorithm for a related largest common subgraph
problem (Cicirello and Smith, 2000). We compared this to several other
standard and custom-built crossover operators and found that it tended to
give the best performance. UPMX works as follows: for two permutations a
and b, for each index i with probability cxswappb, swap the value of ai and
bi . Unless ai = bi , the new values of ai and bi after the swap will occur
in their respective arrays in two places. To fix this, we also swap aj and
bk , where j and k are the indices of the duplicate values in their respective
arrays. An example of this swap process for index i = 0 follows.
1. Initial arrays: a = {1, 3, 2} and b = {2, 1, 3}.
2. Swap the first elements of a and b: a = {2, 3, 2} and b = {1, 1, 3}.
3. a has a duplicate at j = 2 and b has a duplicate at k = 1. Swap aj and
bk : a = {2, 3, 1} and b = {1, 2, 3}.
For mutation, we also adopt a simple swap-based scheme. For a
permutation a, for each index i with probability mutswappb, we randomly
select an index j 6= i and swap ai with aj .
We also make use of two variants on a simple hill climbing algorithm
based on swaps. The first variant, hillClimbN onDominated, performs
repeated random swaps, undoing a swap if it worsens any of the objective
functions. The second variant, hillClimbOneObj, takes an objective name
as an argument, and only undoes a swap if the given objective is worsened,
ignoring any worsening of other objectives. We use the former hill climbing
variant as a core function to improve our population of alignments, and use
the latter to increase the variance of our population in objective space, which
helps us to find a better set of representatives of the approximated Pareto
front.
It is important to note that our crossover, mutation, and hill climbing
functions all use swapping as their basic operation. Since the only way
permutations are modified is by swapping, this allows us to implement our
fitness evaluations very efficiently– we only compute the change in the userspecified objectives at each swap. This allows us to evaluate millions of
swaps per second on a desktop CPU.
Our population is stored in an archive data structure that automatically
discards dominated solutions when better solutions inserted. This gives us
a data structure that stores only the non-dominated solutions that have been
found so far. When the archive exceeds its user-defined size limit, it shrinks
itself to the maximum size by discarding the most crowded solutions, using
the crowded comparison operator first introduced in NSGA-II (Deb et al.,
2002). This helps to ensure that a diverse variety of solutions are maintained
in the population at all times. This archiving strategy is similar to the
multiobjective local search algorithm PAES (Knowles and Corne, 2000).
To increase the performance of our algorithm, it was designed from
the beginning to be amenable to parallelization. The user specifies how
many threads the program is to use, and each thread executes Algorithm
1 until a user-specified time limit has been reached. The threads all
share the same archive, which is the only point of communication
between the threads. This solution is highly scalable, and is able to
fully utilize 16 processor cores in our tests. The overall structure of the
algorithm is very straightforward, and reminiscent of other hybrid genetic
algorithms in the literature; e.g. (Nguyen et al., 2007). One quirk of
the algorithm is that we first decide probabilistically whether to perform
hillclimbOneObj or hillClimbN onDominated, but then afterwards,
perform another round of hillClimbN onDominated. This ensures that,
when hillclimbOneObj is chosen, the alignment is not left in a dominated
state that could easily be improved to a non-dominated one.
Algorithm 1 Per-Thread Loop
while time limit not reached do
(parent1, parent2)
two random members of Archive
child
initialize new alignment
with probability cxrate do
child
crossover(cxswappb, parent1, parent2)
end
with probability mutrate do
child
mutate(mutswappb, child)
end
with probability oneobjrate do
randObj
randomly-chosen objective
child
hillclimbOneObj(randObj, niters, child)
else do
child
hillclimbN onDominated(niters, child)
end
child
hillclimbN onDominated(niters, child)
insert(child, Archive)
The user-adjustable parameters of our algorithm include the time limit,
the rate of crossover (cxrate) and mutation (mutrate), the probability of
performing a swap at each index for crossover (cxswappb) and mutation
(mutswappb), the probability that single-objective hill climbing will be
used instead of non-dominated hill climbing (oneobjrate), and the number
of iterations of hill climbing to perform in each loop (niters). We find that
over the course of the algorithm’s execution, the optimal rate of crossover,
mutation, and one-objective hill climbing seem to vary. We thus provide
the option of automatically adjusting those rates to their rates of success
at producing non-dominated solutions over the course of the algorithm’s
execution. Other, more sophisticated control schemes exist in the genetic
algorithms literature (Eiben et al., 1999; Eiben and Smit, 2011), but this
simple heuristic appears to work quite well. We find that cxswappb = 0.5,
mutswappb = 0.0001, and hillclimbiters = 10000 are good settings
for the remaining parameters. It is important to emphasize that we set the
number of hill climbing iterations quite high, so that computation time is
dominated by hill climbing. We find that hill climbing is best used as the
primary means for obtaining results, with crossover and mutation mostly
playing a secondary role to prevent premature convergence.
3
RESULTS AND DISCUSSION
As discussed above, we report our results aligning the networks
included in Isobase (Park et al., 2011), following the same
methodology as our previous paper comparing existing algorithms,
except that we replace ICS with S 3 (Clark and Kalita, 2014). For
each alignment problem, we run our algorithm for 12 hours. We
try two different sets of objectives. In the first experiment, we set
Optnetalign to maximize S 3 as well as the sum of the BLAST bit
scores of proteins that have been aligned. In the second experiment,
we optimize S 3 and GOC. The first experiment is most directly
comparable to existing aligners, which try to optimize the sum of
the bit score of all pairs of nodes that have been aligned (i.e., a
maximum-weight matching), while the second is an investigation
of how well we can optimize the overlap of GO terms directly,
which helps us obtain an approximate upper bound on the extent
to which both topological fit and biological function can be jointly
optimized. For each alignment problem, Optnetalign is set to output
an approximate Pareto front containing at most 200 alignments.
Since Optnetalign outputs more candidate alignments than the
other aligners we evaluate against combined, we report the average,
minimum, and maximum of S 3 and GOC of the output alignments
for each problem. We also compare the Pareto front of alignments
our algorithm produces to all the alignments produced by the
aligners that have a user-configurable tradeoff parameter between
topological and biological similarity for the C. elegans - D.
melanogaster alignment problem.
We also make some changes in the aligners we compare to, when
compared to our previous benchmark paper. Since our alignment
algorithm is somewhat similar to PISwap and MAGNA, we add
these two aligners to the evaluation. We also remove IsoRank
(Singh et al., 2008), GRAAL (Kuchaiev et al., 2010), MI-GRAAL
(Kuchaiev and Przulj, 2011), and NATALIE (El-Kebir et al., 2011)
from consideration, since they were Pareto dominated by other
existing aligners on this dataset.
To ensure that GO annotations transferred by sequence similarity
were not skewing our results, we also evaluated using only
experimentally-verified GO terms. This did not substantially change
the relative performance of the aligners benchmarked, so we use
all GO terms in the results reported here. For results with only
GO terms with experimental evidence codes, see our supplementary
information file.
3.1
First Experiment Benchmark Results
Here we report the results of optimizing S 3 and the sum of
bitscores of aligned pairs. We report the same data in tabular
form in our supplementary information file. Optnetalign matches or
outperforms all existing aligners on S 3 , and outperforms all but one
aligner on GOC. As noted above, we report the minimum, average,
and maximum of these scores. These metrics are almost perfectly
inversely correlated, so for all of these experiments, the alignment
Optnetalign produces with the highest S 3 has the lowest GOC, and
vice versa.
We first consider our performance on S 3 in Figure 1. We must
note that several aligners crashed on several experiments. This is
denoted by a missing bar in these instances. The maximum S 3
found by Optnetalign on each problem instance is comparable to or
exceeds that of NETAL, which was previously the best performer on
this data set. However, as shown in Figure 2, our topologically best
alignments have much higher GOC scores than NETAL’s. Even
our alignments with the poorest topological quality tend to perform
comparably to or better than many existing aligners.
With respect to GOC scores, our best alignments are all a distant
second place behind PISwap. However, because PISwap works by
first maximizing sequence similarity with the Hungarian algorithm,
and then is limited to performing swaps that don’t worsen the
sequence similarity matching, its topological fit is extremely low,
with S 3 scores generally an order of magnitude lower than any other
aligner. Thus, PISwap’s alignments constitute an extreme point on
the Pareto front. On its topologically worst performance, PISwap
only finds 91 conserved edges on the C. elegans to S. cerevisiae
alignment problem, out of the 4,495 possible. Our best GOC
scores, which outperform all aligners except PISwap, come from
alignments that have non-trivial S 3 scores that are in some cases,
5
C. Clark and J. Kalita
Fig. 1. S 3 score for various aligners and our own for the Isobase dataset,
when Optnetalign is set to optimize S 3 and sum of bit scores. Note that the
bars in the chart are in the same order as in the legend.
Fig. 3. Pareto fronts found by various aligners with adjustable tradeoff
parameters and our own for the C. elegans to D. melanogaster alignment
problem, when Optnetalign is set to optimize S 3 and sum of bit scores.
The scatter plot here is visually indistinguishable from equivalent plots using
other networks from this dataset. We use C. elegans to D. Melanogaster in
this figure because SPINAL and GHOST tended to crash on other problems,
giving incomplete data.
in aligned pairs is comparable to the diversity in all the alignments
produced by previous aligners.
3.2
Fig. 2. GOC for various aligners and our own for the Isobase dataset, when
Optnetalign is set to optimize S 3 and sum of bit scores.
such as the S. cerevisiae to H. sapiens problem, competitive with
existing algorithms.
We also compare the Pareto front found by Optnetalign on the
C. elegans to D. melanogaster problem to the range of tradeoffs
produced by existing aligners that expose a tradeoff parameter to
the user for balancing topological and biological similarity. The
results in Figure 3 show that Optnetalign is able to find a much
wider Pareto front than all previous aligners. Previously, it was
not nearly as clear that such a wide range of results was possible.
The alignments Optnetalign produced for this problem also varied
greatly in the pairs of nodes they chose to align. The least similar
pair of networks had only 8.7% of their aligned pairs in common.
The maximum was 73%, and the average was 36%. This diversity
6
Second Experiment Benchmark Results
Here we report the results we obtain when we set Optnetalign to
optimize S 3 and GOC directly. Since ours is the only aligner among
those tested that can optimize GO term consistency instead of bit
score sum, all other aligners are still optimizing bit score sum, when
applicable, so all of the scores in Figures 4 and 5 are the same as in
Figures 1 and 2, except for Optnetalign’s.
In Figure 4, we see that our S 3 results are much the same as
before with the best alignments, with only slightly lower scores. The
average scores are somewhat lower now, especially on the latter 3
alignments, where the average no longer outperforms most aligners.
The minimum scores are very similar to those in Figure 1, as well.
The primary difference when optimizing GOC directly, instead
of its imperfect proxy of bit score sum, is that we are now able to
obtain GOC scores that outperform PISwap on several problems.
While on the C. elegans to D. melanogaster problem, PISwap still
strongly outperforms all other aligners, Optnetalign is actually able
to outperform PISwap on the latter three alignment problems, and
performs much more strongly on the other problems involving C.
elegans.
3.3
Synthetic Benchmark Results
While the preceding benchmarks show that Optnetalign obtains
excellent performance on real-world data, the true alignment for
such networks is unknown. Therefore, we additionally benchmark
against the NAPAbench network alignment benchmark dataset
(Sahraeian and Yoon, 2012) to evaluate node correctness– the
share of nodes in the smaller network that have been correctly
ICS S 3
EC
GOC
Bit score sum LCCS size
ICS
1.0 -0.132 -0.338 0.079 -0.103
-0.42
S3
1.0
0.957 -0.964 -0.848
0.939
EC
1.0
-0.948 -0.796
0.987
GOC
1.0
0.81
-0.91
Bit score sum
1.0
-0.759
LCCS size
1.0
Table 1. Correlation matrix for various objectives on the S. cerevisiae to
H. sapiens alignment problem. Unlike the other objectives, largest common
connected subgraph (LCCS) size was not set as an optimization objective in
our program; it was computed afterwards.
3.4
Fig. 4. S 3 score for various aligners and our own for the Isobase dataset,
when Optnetalign is set to optimize S 3 and GOC. Note that the bars in the
chart are in the same order as in the legend.
Fig. 5. GOC for various aligners and our own for the Isobase dataset, when
Optnetalign is set to optimize S 3 and sum of GOC.
mapped to orthologous nodes in the second. On these synthetic tests,
Optnetalign achieves performance similar to, but slightly worse
than, the best-performing existing aligners on this dataset, PISwap
and GHOST, with Optnetalign’s best-scoring alignments achieving
node correctness levels between about 77 and 80%. On this dataset,
the alignments with highest node correctness are the alignments
on the high topological side of the Pareto front found; biological
similarity is less helpful, though still necessary, to create correct
alignments. For more details, see our supplementary information
file.
Comparing Objectives
The speed of Optnetalign, its ability to handle any number of
objectives, and the large number of diverse alignments it can create
in a reasonable amount of time, allow us to perform novel analyses
that explore the inherent tradeoffs between many objectives, as
well as examine how different objectives are correlated. This can
give us insight into the nature of the network alignment problem,
and help us identify deficiencies in current alignment evaluation
metrics. We demonstrate this by performing another run of the S.
cerevisiae to H. sapiens problem, in which we set the aligner to
optimize EC, ICS, S 3 , sum of bit scores, and GOC. This time,
we set the maximum archive size very high, to allow the aligner
to output as many non-dominated alignments as possible in a 12
hour run. This resulted in 571 unique, non-dominated alignments.
The minimum agreement on aligned pairs for these alignments was
1%, the maximum was 92%, and the average was 8.3%. This
lack of agreement is consistent with the lack of agreement between
existing aligners (Patro and Kingsford, 2012), so this confirms that
our aligner produces alignments as diverse as using many different
previously-published aligners together.
We present a correlation matrix for our objectives in Table 1. EC
and S 3 are strongly positively correlated, but, as we noted above,
since Optnetalign can minimize the ICS denominator without
increasing the number of conserved edges, ICS is negatively
correlated with the other two topological metrics. We also include
the commonly used metric of largest connected common subgraph
(LCCS) here (see e.g. Kuchaiev and Przulj (2011) for more
information), which Optnetalign cannot optimize directly. To
demonstrate how Optnetalign allows us to compare the behavior
of different alignment metrics, the relationship between ICS and
GOC is illustrated in Figure 3 of our supplementary information.
To better understand ICS inflation, we also present a plot of EC
against ICS in Figure 4 of our supplementary information.
This experiment also demonstrates the potential utility of
optimizing for GO term consistency instead of bit score sum. In
Figure 6 in our supplementary information, we show that, while
bit score sum and GOC generally correlate, the highest GOC
scores we obtain have bit scores sums that are less than half of
the highest bit score sum Optnetalign finds. This reinforces the fact
that, in some cases, proteins of similar function can have significant
differences in their sequences, and it shows the value of optimizing
GOC directly.
This large set of alignments again highlights the inherent conflict
between optimizing measures of topological fit and measures of
biological fit simultaneously. In Figure 5 of our supplementary
7
C. Clark and J. Kalita
information, we show S 3 plotted against GOC. Even though
most of the alignments dramatically outperform those of previous
aligners, we still see a distinct tradeoff between S 3 and GOC.
Since previous aligners varied dramatically in performance while
only outputting a single alignment, there was very little evidence
that such a wide Pareto front between these two objectives was
obtainable. Previous aligners which provided a user-controlled
tradeoff parameter between these topological and biological
similarity scores framed the parameter as a small tweak, and in most
cases, adjusting that parameter through its entire range, as we did in
producing Figure 3, explores only a small portion of objective space.
Therefore, it was not clearly understood that these two objectives
conflict so sharply.
3.5
Potential Directions for Future Work
Our results here show some weaknesses in existing network
alignment research that open up many possibilities for future
work. Perhaps the most glaring question, given these results, is
that of the tradeoff between biological and topological similarity.
The assumption underlying the entire premise of global network
alignment is that different species share large regions of their PPI
networks, which they inherited from a common ancestor. Our results
show that finding that overlap is not as easy as was previously
thought. We can either align two networks in a way that matches
many neighbors in G1 to many neighbors in G2 , or we can
align two networks in such a way that proteins with similar GO
annotations are matched together. It does not appear from our
results that it is possible to do both well. There are a number of
possible reasons for this that should be better explored in future
work. First, as we noted previously (Clark and Kalita, 2014), this
tradeoff is much less dramatic in synthetic networks. It’s possible
that currently known protein networks have too many false positive
and false negative edges to be aligned reliably, and the false positive
edges mislead alignment algorithms. If this is the case, we should
probably prefer alignments with higher GOC, even if they have a
lower S 3 . Second, it is possible that our current best topological
metrics are insufficient and mislead Optnetalign. These metrics all
essentially count how many neighbors in G1 have been matched to
neighbors in G2 , but they have no way of ensuring neighbors of
neighbors are matched to neighbors of neighbors, or that other large
structures between networks are matched properly. One proposed
metric that partially overcomes this is to report the size the largest
connected common subgraph, but we have found that this metric
correlates so strongly with metrics like S 3 that it does not provide
us with much additional helpful information. This makes sense
inuitively: conserving one more edge could join two previously
disconnected common subgraphs and essentially double the score.
Equivalently, reporting the size of the largest common connected
subgraph doesn’t tell us anything about the distribution of the sizes
of the remaining connected subgraphs found. A few moderately
large ones may be much more helpful than one very large one,
especially if the species are more distantly related. Our algorithm
performs well even when optimizing many objectives in one run,
and, as we show above, this can be used to critically evaluate
different candidate objectives.
It would also be possible to use this aligner to evaluate
existing aligners’ approximate topological similarity metrics, such
as GHOST’s (Patro and Kingsford, 2012) spectral graph signatures
8
or graphlet degree signatures (Pržulj, 2007; Milenkovic and Pržulj,
2008). The relative performance of these similarity metrics has
always been unclear, though some preliminary work has been done
on the problem (Milenković et al., 2013; Crawford et al., 2014).
With metaheuristic multiobjective alignment, we could optimize
many of these metrics at once and analyze many alignments to
determine which ones best predict or conflict with the objectives of
network alignment. It’s even possible that some of these similarity
measures, when paired with Optnetalign, could produce better
results than optimizing S 3 or GOC directly.
An important direction for future work is to find ways to
summarize or condense the large number of possible alignments
between two PPI networks. The original goal of pairwise global
alignment was to produce the one best alignment given our
objectives. Our work here has shown that currently studied
alignment objectives do not uniquely specify a best alignment.
Instead, there are many diverse alignments along a wide Pareto
front, and these alignments may agree on as few as 1% of their
aligned pairs while still appearing good with respect to at least one
objective. While this was already known, our multiobjective aligner
makes this problem much more obvious. This creates difficulties for
a biologist who would like to use the output of such an algorithm.
An obvious next step would be to find ways to summarize the
results of these alignments so that someone who wants to use an
alignment can either pick the alignment that best fits their particular
needs, or produce a composite alignment that summarizes the
most interesting alignments found. We have done some preliminary
experiments with creating partial alignments, using the number of
aligned pairs of the alignment as an objective. This could be used
to produce small, local alignments and large, global alignments in
one run of the program as well, which would unify the two related,
but heretofore separate, techniques of local and global network
alignment. This has similarities to NetAligner (Pache and Aloy,
2012; Pache et al., 2012), which is another tool capable of both
local and global alignment. There are many other possibilities for
producing summary or approximate alignments, as well. The recent
publication of DualAligner (Seah et al., 2014) is a good step in this
direction.
4
CONCLUSION
We have shown that Optnetalign can outperform or perform
competitively with the best previously published aligners. Furthermore,
since it is a multiobjective algorithm, it manages to better balance
the tradeoff between the conflicting objectives of biological and
topological fit. Its ability to produce a large number of alignments in
one run allows us to explore the problem of network alignment more
thoroughly, and reveals the strengths and weaknesses of current
evaluation metrics. We expect Optnetalign will be useful not only
in aligning particular PPI networks to one another, but also in
evaluating and experimenting with new alignment objectives, as
well. The large number of highly diverse alignments Optnetalign
can produce allows us to thoroughly understand the sometimes
complex relationship between alignment objectives, and also shows
promise in developing new techniques to combine many alignments
into summary alignments, to produce many-to-many alignments, or
to create local alignments that find small-scale regions of similarity
between PPI networks. Perhaps most importantly, instead of running
many different programs dozens of times to get a wide variety of
possible alignments to analyze, users of alignment software need
only run Optnetalign once.
ACKNOWLEDGEMENT
We thank Vikram Saraph for helping us get MAGNA working,
Cheng-Yu Ma for quickly diagnosing and fixing a bug in PISwap
that we were having difficulty with, and Rob Patro for insightful
discussion.
Funding: This work was supported by a grant from the Biofrontiers
Center at the University of Colorado, Colorado Springs.
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9
Supplementary Information for A Multiobjective Memetic
Algorithm for PPI Network Alignment
Connor Clark and Jugal Kalita
1 NAPAbench synthetic benchmarks
Here, we present bar charts for benchmarks we performed on the NAPAbench synthetic dataset.
Node correctness of various aligners is reported in Figure 1 and S 3 score is reported in Figure 2.
We report the number of edges and nodes in each graph of this dataset in Table 1. Most aligners
perform quite similarly with respect to node correctness on this data, with the exception of the
two topology-only aligners evaluated, NETAL and MAGNA, which perform quite poorly. This
reflects the results in our previous benchmark paper when using this data set (Clark and Kalita,
2014).
NAPAbench includes synthetic networks generated from three distinct network evolution
models: crystal growth (CG), duplication-mutation-complementation (DMC), and duplication
with random mutation (DMR). We run each aligner on three alignment problems from each
network model, and report their average performance on each model, as well as their average
performance overall. Since Optnetalign produces hundreds of alignments for the NAPAbench
problems in one run, we report the average high and low performance, as well. We find that
Optnetalign’s alignments with the highest node correctness have higher S 3 and lower levels of
sequence similarity. However, at the very highest levels of S 3 , node correctness tends to decrease,
suggesting that it is possible to “overfit” when using S 3 as a proxy for node correctness. Studying
this phenomenon further would be an interesting topic for future research.
Alignment Experiment
Crystal Growth 1
Crystal Growth 2
Crystal Growth 3
Duplication-Mutation-Complementation 1
Duplication-Mutation-Complementation 2
Duplication-Mutation-Complementation 3
Duplication with Random Mutation 1
Duplication with Random Mutation 2
Duplication with Random Mutation 3
G1 Edges, G2 Edges
(11986, 15986)
(11987, 15987)
(11984, 15984)
(6090, 8112)
(6361, 8310)
(6077, 8347)
(6017, 8238)
(5739, 7602)
(6457, 8431)
Table 1: Number of edges in the NAPAbench graphs. Each G1 had 3000 nodes, and each G2
had 4000 nodes.
1
Figure 1: Node correctness of benchmarked aligners on the NAPAbench data set. We report
the average node correctness for each of the three network evolution models in the benchmark
set, as well as the average over all models. Similarly to the Isobase tests, since Optnetalign
produces many alignments, we report its average high and low scores as well.
Figure 2: S 3 measure of benchmarked aligners on the NAPAbench data set.
2
2 Isobase networks
The number of nodes and edges for the networks in Isobase that we used are given in the table
below.
Network
C. Elegans
D. Melanogaster
H. Sapiens
S. Cerevisiae
Nodes
2805
7518
9633
5499
Edges
4495
25635
34327
31261
3 Isobase benchmark results as tables
3.1 S 3 Scores
Aligner
NETAL
PINALOG
SPINAL mode 1
SPINAL mode 2
C-GRAAL
GHOST
PISwap
MAGNA
Optnetalign Exp 1 Min
Optnetalign Exp 1 Avg
Optnetalign Exp 1 Max
Optnetalign Exp 2 Min
Optnetalign Exp 2 Avg
Optnetalign Exp 2 Max
ce-dm
0.4117
6.518e-2
crashed
0.1395
0.1429
0.1392
8.7976e-3
0.2611
0.1284
0.2095
0.3960
0.1222
0.2193
0.3796
ce-hs
0.3963
6.4718e-2
crashed
0.1363
0.1306
0.1482
1.07616e-2
0.2447
0.1164
0.2138
0.3963
0.1324
0.2276
0.3699
ce-dm
12.0192
(229.8318
0.0
79.5726
75.957
181.4609
1428.7772
12.9672
84.37
196.95
245.11
305.1
575.2
697.5
ce-hs
18.3437
96.6236
0.0
43.3977
41.215
84.8546
669.9611
18.6436
38.39
84.29
104.61
219.3
381.8
467.7
ce-sc
0.3234
5.3556e-2
3.36515e-2
9.8203e-2
9.0225e-2
0.1186
4.7896e-3
0.2457
0.1502
0.2407
0.3677
0.12
0.185
0.356
dm-hs
0.1873
5.0076e-2
3.1915e-2
0.1234
0.1186
0.1151
1.3216e-2
0.1067
0.07821
0.14329
0.19942
0.06686
0.1165
0.1886
sc-dm
0.1586
4.7737e-2
3.4038e-2
0.109
0.1071
crashed
6.8586e-3
0.1071
0.08615
0.1233
0.1635
0.0318
0.0675
0.1606
sc-hs
0.1728
7.003e-2
4.5275e-2
0.1077
0.1096
crashed
2.6289e-2
0.1163
0.1185
0.1469
0.203
0.031
0.067
0.1857
3.2 GOC Scores
Aligner
NETAL
PINALOG
SPINAL mode 1
SPINAL mode 2
C-GRAAL
GHOST
PISwap
MAGNA
Optnetalign Exp 1 Min
Optnetalign Exp 1 Avg
Optnetalign Exp 1 Max
Optnetalign Exp 2 Min
Optnetalign Exp 2 Avg
Optnetalign Exp 2 Max
3
ce-sc
26.4369
136.5004
145.3108
60.0272
53.48089
95.4396
504.0438
25.2659
48.38
117.24
158.56
233
392.7
435.6
dm-hs
39.7137
302.3266
307.7264
121.2673
110.4886
166.8659
718.5602
37.5428
90.09
211.61
323.11
377.1
791.8
1018
sc-dm
48.41
364.17
375.88
113.216
100.54
crashed
518.55
48.85
64.31
257.13
399.73
251.8
850.1
1030.5
sc-hs
85.627
319.8
335.99
144.5
123.208
crashed
460.04
79.59
159.7
291.5
352.6
386.1
1029.4
1180.9
4 Example Pareto Front
Here, we show the Pareto front our aligner produces in Experiment One for the C. elegans to
D. Melanogaster problem, compared to the alignments created by various other aligners. To
account for PISwap’s extremely high GOC, we use a log scale. The scatter plots for the other
problems look virtually identical to this one, so we only include one example.
1000
AlignerName
CGRAAL
GHOST
GOC
MAGNA
NETAL
optnetalign
100
PINALOG
PISwap
SPINALMode2
10
0.0
0.1
0.2
0.3
0.4
3
S
5 Additional Scatter Plots
Here we present additional scatter plots from our analysis of the interaction and tradeo↵ between
di↵erent alignment objectives.
4
GOC
1000
750
500
0.25
0.50
ICS
0.75
1.00
Figure 3: Scatter plot of ICS scores vs. GOC scores for non-dominated solutions on the S.
cerevisiae to H. sapiens alignment problem when optimizing EC, ICS, S 3 , sum of bitscores,
and GOC.
5
0.25
EC
0.20
0.15
0.10
0.05
0.25
0.50
ICS
0.75
1.00
Figure 4: Scatter plot of ICS vs. EC for non-dominated solutions on the S. cerevisiae to H.
sapiens alignment problem when optimizing EC, ICS, S 3 , sum of bitscores, and GOC.
6
0.8
LCSC
0.6
0.4
0.2
0.0
0.05
0.10
0.15
3
S
Figure 5: Scatter plot of S 3 vs. largest connected shared component (LCSC, also known as
largest connected common subgraph) for non-dominated solutions on the S. cerevisiae to H.
sapiens alignment problem when optimizing EC, ICS, S 3 , sum of bitscores, and GOC.
7
Figure 6: Scatter plot of GOC scores vs. sum of bit scores for non-dominated solutions on the
S. cerevisiae to H. sapiens alignment problem when optimizing EC, ICS, S 3 , sum of bitscores,
and GOC.
8
Figure 7: Scatter plot of S 3 scores vs. GOC scores for non-dominated solutions on the S.
cerevisiae to H. sapiens alignment problem when optimizing EC, ICS, S 3 , sum of bitscores,
and GOC.
9
6 GO terms with experimental evidence only
In this section, we present the GOC bar charts for experiments 1 and 2 of the main paper,
when only GO terms with experimental evidence codes are used. For experiment 1, when we
set our aligner to optimize bit score sum and S 3 , this allows us to verify that our alignments
are meaningfully finding good alignments of GO terms that were not assigned simply on the
basis of sequence similarity. As in several previous papers (Aladag and Erten, 2013; Clark and
Kalita, 2014), we find that, apart from making the absolute GOC scores lower for all aligners,
this does not change their relative performance on the GOC metric. For this reason, we include
these results in this supplementary information file only. Experiment 1 results are shown in
Figure 8 and experiment 2 results are shown in Figure 9.
10
Figure 8: Experiment 1 GOC scores for all aligners evaluated, when using only GO terms with
experimental evidence annotations. The relative performance of aligners is unchanged when
compared to the analogous figure in our paper.
Figure 9: Experiment 2 GOC scores for all aligners evaluated, when using only GO terms with
experimental evidence annotations. The relative performance of aligners is unchanged when
compared to the analogous figure in our paper. However, since our aligner uses GO terms
directly in this experiment, it is able to obtain more robust performance than other aligners
because its alignments are then guided by the restricted set of GO terms.
11
7 Execution Times
The benchmarked aligners di↵ered greatly in their execution times. Unfortunately, since not
all aligners were available for the same platform, we had to execute many of them on virtual
machines, which prevented us from performing a precise comparison of their relative execution
times. Instead, we report here “ballpark” execution times, expressed in orders of magnitude.
Like several of the best-performing aligners, Optnetalign requires several hours to obtain good
results. Since Optnetalign is a metaheuristic, it can conceivably keep trying to improve its
alignments indefinitely. However, after between 10 to 12 hours, it usually failed to find any
further improvement. For this reason, we ran all tests of Optnetalign with a time limit of 12
hours. Comparison here is not entirely straightforward, however, since Optnetalign produces
hundreds of alignments in this amount of time. In contrast, other aligners with adjustable
tradeo↵ parameters must be run again to produce several alignments at di↵erent tradeo↵s. In
such situations, Optnetalign can save the user a significant amount of time.
Aligner
NETAL
PINALOG
SPINAL
GRAAL
C-GRAAL
GHOST
IsoRank
PISwap
MAGNA
Optnetalign
Time
seconds
minutes
minutes
hours to
hours to
hours to
minutes
minutes
hours to
hours to
days
days
days
days
days
8 Execution arguments and parameters
Here we report the particular settings and parameters we used for each aligner. We made
every e↵ort to use the settings recommended in the documentation for these aligners, or the
settings reported to be best in the original publication of these aligners. When no best settings
were reported, we used the settings given in examples in the documentation. In some cases,
noted below, we had to adjust these settings because of program crashes or time constraints.
For details on the meanings of these parameters, refer to the corresponding publication and
documentation.
12
Aligner
NETAL
PINALOG
SPINAL mode 1 and 2
GRAAL
C-GRAAL
GHOST
IsoRank
PISwap
MAGNA
Parameters
-a 0.0001 -b 0 -c 0.5 -i 2
No user-provided parameters.
alpha = 0.7
alpha = 0.8
No user-provided parameters.
For the extractor, k = 3, except for H. Sapiens, where the
extractor ran out of memory,
in which case we used k = 2.
For the aligner, matcher: linear, nneighbors: all, beta: 1.0,
ratio:8.0, searchiter: 10.
–K 30 –thresh 1e-4 –alpha 0.9
–maxveclen 1000000
We used the provided example
Python script, which has no
parameters.
-m S3 -p 15000 -n 2000
Bibliography
Aladag, A. E. and Erten, C. (2013). SPINAL: scalable protein interaction network alignment. Bioinformatics, 29(7), 917–924.
Clark, C. and Kalita, J. (2014). A comparison of algorithms for the pairwise alignment of biological networks. Bioinformatics,
30(16), 2351–2359.
13